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Ground State Solutions For Local-Nonlocal Shrodinger Equations In the Presence Of Two Critical Exponents

Yu Su, Hichem Hajaiej, Hongxia Shi

Abstract

In this paper, we address the existence of ground state solutions for Schrodinger equations in the presence of local and nonlocal operators and two critical nonlinearities associated with each operator. The situation is completely solved in the critical case both in the local and in the nonlocal settings. However, methods developed in these cases cannot extend to local-nonlocal operators when we have two critical power nonlinearities. This unprecedented situation in PDEs presents some challenges, and its resolution will open the door to solve similar problems. Our approach is essentially based on a subtle generalization of the Lieb translation theorem. We will also discuss the positivity of the ground states as well as their regularity.

Ground State Solutions For Local-Nonlocal Shrodinger Equations In the Presence Of Two Critical Exponents

Abstract

In this paper, we address the existence of ground state solutions for Schrodinger equations in the presence of local and nonlocal operators and two critical nonlinearities associated with each operator. The situation is completely solved in the critical case both in the local and in the nonlocal settings. However, methods developed in these cases cannot extend to local-nonlocal operators when we have two critical power nonlinearities. This unprecedented situation in PDEs presents some challenges, and its resolution will open the door to solve similar problems. Our approach is essentially based on a subtle generalization of the Lieb translation theorem. We will also discuss the positivity of the ground states as well as their regularity.
Paper Structure (7 sections, 20 theorems, 176 equations)

This paper contains 7 sections, 20 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Sobolev Space
  3. Lieb's translation theorem
  4. Generalized version of Lieb's translation theorem
  5. Mountain pass geometry and Nehari manifold
  6. Proof of Theorem \ref{['Theorem1.2']}
  7. $L^{\infty}$ estimate

Key Result

Theorem 1.1

BaoQQ2024MalaysianChergui-Gou-Hajaiej2023CVPDE Let $N\geqslant3$ and $0<s<1$. Then we have the following results.

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 3.1
  • ...and 29 more